Sum rules

First, prepare the matrix using the simple factorization.

    α1,β1,γ1 = rand(Float64,3)
    xx=bloquesun(4,1,(α1,β1,γ1))
    α2,β2 = rand(Float64,2)
    yy=bloquesun(4,2,(α2,β2,α2))
    α3,β3,γ3 = rand(Float64,3)
    zz=bloquesun(4,1,(α3,β3,γ3))
    α4,β4 = rand(Float64,3)
    xx2=bloquesun(4,3,(α4,β4,α4))
    α5,β5 = rand(Float64,2)
    yy2=bloquesun(4,2,(α5,β5,α5))
    α6,β6,γ6 = rand(Float64,3)
    zz2=bloquesun(4,1,(α6,β6,γ6))

    mat4 = xx*yy*zz
    mat4c1 = xx*yy*zz*xx2*yy2

Then obtain the basis states:

    welcome = basis_states([2,0,0,0])

Finally, compute the rate

    rate1 = abs( group_function([2,0,0,0], welcome[9], welcome[9], mat4) )^2 + abs( group_function([2,0,0,0], welcome[9], welcome[4], mat4) )^2 + abs( group_function([2,0,0,0], welcome[9], welcome[7], mat4) )^2
    rate2 = abs( group_function([2,0,0,0], welcome[9], welcome[9], mat4c1) )^2 + abs( group_function([2,0,0,0], welcome[9], welcome[4], mat4c1) )^2 + abs( group_function([2,0,0,0], welcome[9], welcome[7], mat4c1) )^2
    rate1 ≈ rate2